y<-c( 5.06, 5.01, 5.12, 5.13, 5.14, 5.16, 5.25, 5.19, 5.24, 5.46,
      5.40, 5.57, 5.47, 5.53, 5.61, 5.59, 5.61, 5.75, 5.68, 5.80)
x<-c( 0.0,0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2,
      2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8)
windows()
plot(x,y)
#funzione di R


#metodo scolastico calcolo statistiche di base
n<-length(y)
n

mx<-mean(x)
mx
my<-mean(y)
my
vy<-var(y)*(n-1)/n
vy
vx<-var(x)*(n-1)/n
vx
mxy<-mean(x*y)
mxy
vxy<-mxy-mx*my
vxy
#stima intercetta e coeff. angolare
b<-vxy/vx
b
a<-my-b*mx
a
#calcolo var residua
r2<-vxy^2/vx/vy
r2
stilde<-vy*(1-r2)*n/(n-2)

stilde
sqrt(stilde)

yhat<-a+b*x
windows()
plot(x,y,type="b")
points(x,yhat,col="red",type="b")
stilde2<-sum((y-yhat)^2)/(n-2)
#inferenza su coeff. angolare H_0: beta1=0.2 alpha=0.01
#alternativa dx
vb<-stilde2/n/vx
test<-(b-0.2)/sqrt(vb)
perc<-qt(0.99,n-2)
pvalue<-1-pt(test,n-2)
perc
test
pvalue
#inferenza su coeff. angolare #IC 0.99##################
##########################################################
vb<-stilde/n/vx
perc<-qt(0.995,n-2)
L<-b-perc*sqrt(vb)
U<-b+perc*sqrt(vb)
c(L,b,U)
#inferenza su intercetta beta0 =5 alternativ bil alpha=0.01
va<-stilde2*(1/n+mx^2/n/vx)
testa<-(a-5)/sqrt(va)
perca<-qt(0.995,n-2)
pvaluea<-2*(1-pt(abs(testa),n-2))
perca
testa
pvaluea
#IC su beta0 0.99##################################
###################################################
va<-stilde*(1/n+mx^2/n/vx)
perca<-qt(0.995,n-2)
L<-a-sqrt(va)*perca
U<-a+sqrt(va)*perca
c(L,a,U)
#inferenza su valore atteso beta_0+beta_1*0.2=5 alt dx alpha=0.05
y_02<-a+b*0.2
v_02<-stilde*(1/n+(0.2-mx)^2/n/vx)
perc<-qt(0.95,n-2)
testy<-(y_02-5)/sqrt(v_02)
pvalueU<-1-pt(testy,n-2)
perc
testy
pvalueU
#intervallo conf illimitato dx per mu= beta_0+beta_1*0.2 alpha=0.05
L<-y_02-perc*sqrt(v_02)
c(L,Inf)
#intervallo confidenza illimitato sx
U<-y_02+perc*sqrt(v_02)
c(-Inf,U)
#intervallo confidenza bilaterale
percbil<-qt(1-0.05/2,n-2)
U<-y_02+percbil*sqrt(v_02)
L<-y_02-percbil*sqrt(v_02)
c(L,U)
####SCOLASTICO PROPEDEUTICO PER CHI USA LA CALCOLATRICE

#creazione tabella lavoro
table<-cbind(x,y,x^2,y^2,x*y,a+b*x,y-a-b*x)
tableT<-rbind(table,colSums(table),colSums(table)/n)

colnames(tableT)<-c("x","y","x^2","y^2","x*y","fitted","res")
tableT
#salvataggio file formato .tex serve library xtable e poi latex per compilarlo
#library(xtable)
#colnames(working.table)<-c("x","y","x^2","y^2","x*y")
#edit(working.table)
#workreg<-xtable(tableT,caption="regressione-tabella di lavoro",digits=4)
#print(workreg,file="regress.tex")
#garfici
#windows()
plot(x,y,type="b")

abline(a=a,b=b,col="red")
#windows()
plot(y-a-b*x,type="b")
abline(a=0,b=0)
#funzione lm
res<-lm(y~x)
summary(res)